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This is my favourite brain teaser, and one of the most difficult that I know. I apologize if it has been posted before and I missed it.

A team of 15 takes part in a contest. They know each other and all the rules of the contest, and they have time to make their strategy before the actual contest starts.

The rules of the contest are:

1. The players sit in a circle, so that everybody sees everybody. Then the players are blindfolded.

2. On each player's head, either a red or a blue cap is placed. The colour of the cap is determined completely arbitrarily, e.g. by coin toss.

3. The blindfold is removed, and everybody has to write on a piece of paper the answer to the question "What is the colour of your cap?". The answer must be "RED", "BLUE" or "---". Everybody has to write down their answer at the same time.

4. Everybody shows the paper at the same time.

5. The players do not cheat. (They do not exchange information through words, utterances, signs, facial expressions, delays in writing their answer, etc.)

The procedure of the contest is done only once. The team wins the contest if at least one answer is correct and no answer is incorrect. "---" is considered a neutral answer, neither right nor wrong.

Examples:

if team member #12 correctly answers RED, and everybody else answers "---", they win.

if all members answer correctly (RED/BLUE), except for member #12 (who said RED and had blue), they lose.

Which is the strategy with the biggest chance to win, and how high is that chance?
Have fun!

[P.S.] For those who know Romanian, this and many other brainteasers are to be found at www.chichitza.com and www.chichitza.com/forum

As for chess, a section in English is www.chichitza.com/chess.html . It contains puzzles and studies. Some studies may be more difficult for computers than for humans

This is my favourite brain teaser, and one of the most difficult that I know. I apologize if it has been posted before and I missed it.

A team of 15 takes part in a contest. They know each other and all the rules of the contest, and they have time to make their strategy before the actual contest starts.

The rules of the contest are:

1. The players sit in a circle, so that everybody sees everybody. Then the players are blindfolded.

2. On each player's head, either a red or a blue cap is placed. The colour of the cap is determined completely arbitrarily, e.g. by coin toss.

3. The blindfold is removed, and everybody has to write on a piece of paper the answer to the question "What is the colour of your cap?". The answer must be "RED", "BLUE" or "---". Everybody has to write down their answer at the same time.

4. Everybody shows the paper at the same time.

5. The players do not cheat. (They do not exchange information through words, utterances, signs, facial expressions, delays in writing their answer, etc.)

The procedure of the contest is done only once. The team wins the contest if at least one answer is correct and no answer is incorrect. "---" is considered a neutral answer, neither right nor wrong.

Examples:

if team member #12 correctly answers RED, and everybody else answers "---", they win.

if all members answer correctly (RED/BLUE), except for member #12 (who said RED and had blue), they lose.

Which is the strategy with the biggest chance to win, and how high is that chance?Have fun!

[P.S.] For those who know Romanian, this and many other brainteasers are to be found at www.chichitza.com and www.chichitza.com/forum

As for chess, a section in English is www.chichitza.com/chess.html . It contains puzzles and studies. Some studies may be more difficult for computers than for humans

Assuming that there is no limitations on number of caps of either color (i.e., they might be wearing all red or all blue), it would be wise to

Spoiler for follow this strategy

Appoint a leader. This leader would check which color is appearing lesser number of times on others heads and would write that color. All others should write '---'.Assumption here is that on a fair coin toss both colors should have same probability of repetition.

Assuming that there is no limitations on number of caps of either color (i.e., they might be wearing all red or all blue), it would be wise to

Spoiler for follow this strategy

Appoint a leader. This leader would check which color is appearing lesser number of times on others heads and would write that color. All others should write '---'.Assumption here is that on a fair coin toss both colors should have same probability of repetition.

I came up with a similar solution except

Spoiler for

they do want to assign one person to make the guess, but it doesn't matter what color the other hats are. Even if the "leader" was looking at all red hats, he would still have a 50/50 chance of having red or blue.

Assuming that there is no limitations on number of caps of either color (i.e., they might be wearing all red or all blue), it would be wise to

Spoiler for follow this strategy

Appoint a leader. This leader would check which color is appearing lesser number of times on others heads and would write that color. All others should write '---'.Assumption here is that on a fair coin toss both colors should have same probability of repetition.

That strategy is not really required:

Spoiler for because

..if each cap is decided with a coin toss, the color of the cap on the leader's head is completely independent from what he/she sees. In other words, even if the leader sees 14 REDs, the probability that the leader will have a RED is still half. So the strategy is not better than a random guess.

have one team member wear mirrored sunglasses and seat across from him/her the team member with the best eyesight. Have all team members answer '---' except for the eagle-eyed member who will look in the sunglasses of the member sitting opposite him/her and write down which color he/she sees in their reflection. 50-100% chance of success depending on distance, eyesight, atmospheric disturbances, temporary blindness, retinal detachment, etc.

Seems that Fast Reply is not working for me. Sorry if this appears multiple times.

Spoiler for I don't think this answer would be considered cheating under the specified rules

Everyone but one select person writes "--". When the masks are removed, everyone looks at the selected person if that person is wearing red and looks at others if blue. Selected person therefore knows what color to write. People are looking at me - Red, otherwise Blue.

have one team member wear mirrored sunglasses and seat across from him/her the team member with the best eyesight. Have all team members answer '---' except for the eagle-eyed member who will look in the sunglasses of the member sitting opposite him/her and write down which color he/she sees in their reflection. 50-100% chance of success depending on distance, eyesight, atmospheric disturbances, temporary blindness, retinal detachment, etc.

Seems that Fast Reply is not working for me. Sorry if this appears multiple times.

Spoiler for I don't think this answer would be considered cheating under the specified rules

Everyone but one select person writes "--". When the masks are removed, everyone looks at the selected person if that person is wearing red and looks at others if blue. Selected person therefore knows what color to write. People are looking at me - Red, otherwise Blue.

Remember:5. The players do not cheat.

To put the problem another way, all players of the team are housed in separate rooms. Everyone is informed about the colour of all others' caps by an arbiter. There is absolutely no possibility of learning more information than those colours.

To put the problem another way, all players of the team are housed in separate rooms. Everyone is informed about the colour of all others' caps by an arbiter. There is absolutely no possibility of learning more information than those colours.

How about this strategy

Spoiler for

Let's make it easy on ourselves and call the state of the hats 0 and 1.

Every participant look and sum up all the hat state. The sum should be between 0 and 14.

If a participant see a sum that is odd, abstain from guessing.

If a participant see a sum that is 0, 2, 4, or 6, guess that his hat state is 1.

If a participant see a sum that is 8, 10, 12, or 14, guess that his hat state is 0.

..if each cap is decided with a coin toss, the color of the cap on the leader's head is completely independent from what he/she sees. In other words, even if the leader sees 14 REDs, the probability that the leader will have a RED is still half. So the strategy is not better than a random guess.

Spoiler for Is it really a random guess?

Provided the user knows information of other 14 hats, wouldn't it be much safer to calculate probability using this information. In other words, what is the probability that both the children are girls when you already know one of them is a girl? Most people say it is 1/3 while few debate it would remain 1/2. For this puzzle I will go with the first option, as otherwise the best strategy will be - all except one predesignated person taking a guess with 1/2 probability. Not much of a strategy if you ask me.